Interfacial Dynamics and Rheology of a Crude-Oil Droplet Oscillating in Water at a High Frequency

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Interfacial Dynamics and Rheology of a Crude-Oil

Droplet Oscillating in Water at a High Frequency

Nicolas Abi Chebel, Antoine Rémy Piedfert, Benjamin Lalanne, Christine

Dalmazzone, Christine Noïk, Olivier Masbernat, Frédéric Risso

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To cite this version:

Abi Chebel, Nicolas

and Piedfert, Antoine Rémy

and Lalanne, Benjamin

and Dalmazzone, Christine and Noïk, Christine and Masbernat, Olivier

and

Risso, Frédéric

Interfacial Dynamics and Rheology of a Crude-Oil Droplet

Oscillating in Water at a High Frequency. (2019) Langmuir, 35 (29). 9441-9455.

ISSN 0743-7463

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Interfacial Dynamics and Rheology of a Crude-Oil Droplet

Oscillating in Water at a High Frequency

Nicolas Abi Chebel,

†,‡,∥

Antoine Piedfert,

†,‡,∥

Benjamin Lalanne,

‡,∥

Christine Dalmazzone,

§

Christine Noïk,

§

Olivier Masbernat,

‡,∥

and Frédéric Risso

*

,†,∥

†_{Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, 31400 Toulouse, France}

‡_{Laboratoire de Génie Chimique (LGC), Université de Toulouse, CNRS, 31432 Toulouse, France}

§_{IFP Energies nouvelles, 1-4 avenue de Bois Préau, 92852 Rueil-Malmaison, France}

∥_{FR FERMAT, Université de Toulouse, CNRS, INPT, INSA, UPS, Toulouse, France}

ABSTRACT: We report investigations of a pendant diluted crude-oil droplet in water that is forced to oscillate at a frequency ω. The droplet interface contains a significant amount of surface-active agents and displays a marked viscoelastic rheology with elastic moduli larger than viscous ones. At a low frequency, fluid viscosity and inertia are negligible, which allows a direct determination of the dilatational interface rheology. At a large frequency, eigenmodes of inertial shape oscillations are excited. By decomposing the interface shape into spherical harmonics, the resonance curves of the inertial modes of the interface are determined, as well as the frequency and damping rate of each mode. These two parameters are of major importance for the prediction of the deformation and breakup of a droplet in any unsteadyflow without any prior knowledge of either the chemical composition or the detailed rheological properties of the interface. Then, interfacial rheology is related to interface dynamics by solving the coupled dynamic equations for the twofluids and the interface. It turns out that the rheology of the interface is well described by an equivalent two-dimensional viscoelastic material, the elasticities and viscosities of which depend upon the frequency. Afirst significant result is that shear and dilatational elasticities are closely connected, as are shear and dilatational viscosities. This implies that intrinsic rheology plays a major role and that compositional rheology is either negligible or strongly coupled to the intrinsic one. A second major result is that, for moderately aged droplets (≤5000 s), the elasticity and viscosity at

a high frequency (10−80 Hz) can be extrapolated from low-frequency measurements (≤1 Hz) by a simple power law of the

frequency,ωz. The exponent z is related to the loss angleθlossby a relation found in many previous low-frequency investigations

of crude-oil interfaces: z =θloss/2π. The present work thus extends classic observations obtained at a low frequency to a higher

frequency range corresponding to the natural frequency of the droplets, where the droplet shape results from the balance between dynamic pressure and surface stresses and the interface involves simultaneous shear and dilatation. These results bring about serious constraints regarding the modeling of physicochemical underlying mechanisms and provide some insights for the understanding of the structure of crude-oil interfaces.

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INTRODUCTION

The original motivation of this work is to predict how a crude oil−water interface deforms and eventually ruptures, which is of special importance for the prediction of droplet size distributions and the stability of petroleum emulsions.

The deformation of a droplet results from the response of the interface to the hydrodynamic stress exerted by the surrounding ﬂuid.1

Provided the ﬂuids are not too viscous, the interface

dynamics is characterized by a series of eigenmodes of

oscillations.2(Low viscosity means that the Ohnesorge number

should be small: Oh= _{ρ γ}η ≪1

d

i i

, where ρi and ηi are

respectively the density and the viscosity of the innerﬂuid, d is the droplet diameter, andγ is the interfacial tension). Each of

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these modes, deﬁned by its order n ≥ 2, is characterized by a speciﬁc shape, an angular frequency ωn, and a damping rateβn.3

A droplet that is initially deformed according to mode n and

released in a ﬂuid at rest performs damped oscillations at

frequencyωnwith an amplitude that exponentially decays with

time at a rateβ_n, until it eventually reaches a spherical shape. These oscillations result from the periodic exchange between the kinetic energy of theﬂuids and the potential energy stored in the deformed interface. Bothωnandβnare increasing functions of n

and decreasing functions of the droplet size, whereas the ratio βn/ωnis a decreasing function of d. In many situations involving

droplet emulsions, the frequencies are large and the damping rates are small compared to the frequencies. Let us consider a droplet of heptane in water: for d = 0.1 mm, Oh = 7× 10−3,ω2/

2π ≈ 7 × 103_Hz,_β

2≈ 4 × 103s−1; for d = 5 mm, Oh = 10−3,ω2/

2π ≈ 20 Hz, β2≈ 4 s−1. Because they are easier to excite, the

lower-order modes are most often suﬃcient to describe the main deformation of a droplet. In particular, it has been shown that the breakup of a droplet in a turbulentﬂow essentially depends on ω₂ and β₂ and on the properties of turbulence.4−6 It is important to stress that these modes involve frequencies of at least several tens of hertz, which implies that the knowledge of the response of the interface at such large frequencies is of primary importance.

In any case, the behavior of a droplet thus crucially depends on the values of the characteristic frequencies and damping rates of the interface. In general, ω_n and β_n are functions of the mechanical properties of both thefluid and the interface. For Newtonianfluids, these properties are the densities of the inner (ρi) and the outer (ρo)fluids, and the viscosities of the inner (ηi)

and the outer (ηo) ﬂuids. For a clean interface between two

immiscible pure ﬂuids, the interface mechanics is fully

characterized by the interfacial tension γ. However, in the

presence of surface-active agents, the mechanical law of the interface can be much more complex and involve surface elasticity and viscosity. Ideally, the determination ofω_nandβ_n for a givenfluid system should involves three successive steps: (1) the chemical characterization of all the species that are present within bulkfluids as well as adsorbed at the interface; (2) the determination of the rheological properties of the interface resulting from the interactions between all these species; (3) the derivation of the interface-shape eigenmodes resulting from the specific mechanical laws of the interface under consideration. In practice, each of these steps is complex and involves a different scientific field: physico-chemistry, interfacial rheology, and fluid dynamics. Deriving one step from the preceding is not an easy task.

Connecting surface rheology (2) to chemical structure (1) shows the existence of two different origins of elasticity and viscosity of the interface.7Thefirst is related to the variations of the surface energy with the chemical composition of the interface8and it can be considered to be of a compositional or a thermodynamic nature. Neglecting transfers of molecules between the bulkfluids and the interface, each increase (resp. decrease) of the interface area is associated with a decrease (resp. increase) of the interfacial surfactant concentration. This gives birth to a surface elasticity, which is usually called Gibbs elasticity. When surface-active molecules have time to adsorb or desorb during the duration of the process of interface dilatation (or compression), the Gibbs elastic modulus depends on the rate of molecule transfer. Moreover, this mass transfer causes fluid flows in the vicinity of the interface which generates viscous dissipation. Even if this dissipation is not located within the

interface, it gives rise to an apparent viscosity of the interface, which must not be confused with a true surface viscosity in the interpretation of experimental results. It is worth noting that compositional surface rheology is of an extrinsic nature as it depends not only on the interface properties but also on the physical properties and dynamics of thefluids that surround the interface. In addition, the corresponding elasticity and viscosity are only related to surface dilatation. The second origin of interfacial rheology is associated with interactions between adsorbed molecules. As the interface is deformed, the work of intermolecular forces under the change of the distance between molecules gives rise to a surface elasticity, whereas the motion of the molecules through the solvent and a possible irreversible reconfiguration of their relative positions and orientations can generate a surface viscosity. As these mechanisms take place within the interface and do not depend on bulk-fluid properties, they are considered to be of an intrinsic or a rheological nature. At variance with compositional rheology, intrinsic rheology is involved in both surface deformation at a constant area and area variation. It thus requires the introduction of a dilatation elastic modulus and a shear elastic modulus, as well as a dilatation viscosity and a shear viscosity.

Given a surface rheology (2), it is in principle possible to determine the interface dynamics (3). This requires to consider

a coupled system of partial diﬀerential equations involving

Navier−Stokes equations describing the dynamics of both bulk ﬂuids, transport of species through the ﬂuids, molecules transfers

between the bulk ﬂuids and the interface, and mechanical

constitutional law of the interface. The interface-shape eigenmodes are found by solving this system after linearization under the assumption of small amplitude oscillations. Expressions for the frequencyωnand the damping rateβnare

in general not explicit but under the form of the cancellation of a matrix determinant. Such solutions have been obtained for droplets constituted of two Newtonianﬂuids separated by a pure interface (γ = cte),3,9,10an interface with intrinsic elasticity and viscosity,3 and an interface with Gibbs elasticity and intrinsic viscosity.11,12

Various experimental methods have been developed to determine the rheological properties of interfaces: oscillating pendant drop,13−15 biconical disk interfacial rheometer,16,17 Langmuir trough with oscillating barriers,18capillary pressure tensiometer,19,20double wall-ring interfacial rheometer,21,22and magnetic rod interfacial stress rheometer.23 In general, the measured values of elasticity and viscosity are observed to depend upon the time scale at which the surface deformation occurs. Most of the time, these devices are operated at low frequency (ω/2π ≲ 1 Hz), in order to facilitate the accounting of bulk hydrodynamic eﬀects in the determination of interfacial properties. Oscillations of a hemispherical bubble were used to investigate rheological interface properties in the range of 1−500 Hz.24−28Despite the high-frequency range considered, because

of the small size of the bubble (d ≈ 0.5 mm), the forcing

frequency was negligible compared to those of the shape

eigenmode (ω₂/2π ≈ 103 _{Hz). The shape thus remained}

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Oscillation frequencies and damping rates were measured for droplets of various solutions of surfactant in water with concentrations below the critical micelle concentration. Rheological properties were determined byﬁtting the measured values ofω₂andβ₂with the theoretical values based on Gibbs elasticity and dilatational viscosity, but without considering intrinsic elasticity.

Crude oil contains many components that can inﬂuence the

rheology of the oil−water interface of droplets, such as

asphaltenes, resins, solid particles, and acids.30Among them, asphaltenes play a major role because of their ability to form an irreversible network at the interface.31An extensive literature has been devoted to low-frequency rheological characterization of

the oil−water interface by considering either model ﬂuids

containing asphaltenes30,32−35or crude oil.30,31,36,37,37−41Most investigations made use of the oscillating pendant drop method,30−33,36−41 which gave access to the dilatational rheological properties, whereas a few works also used a double wall-ring interfacial rheometer, which measured shear proper-ties.34,35It turns out that oil−water interfaces display viscoelastic properties that depend on many parameters such as oil composition, chemical structure of asphaltenes, temperature, acidity, surface coverage (related to surface aging), and time scale (or frequency) of the interface deformation. Young interfaces with low surface coverage are expected to be dominated by compositional rheology as oscillating drop

measurements are found31 in agreement with the Lucassen−

van den Tempel (LVDT)42model. Aged interfaces rather imply

intrinsic elasticity.

Regarding low-frequency dilatational rheology based on oscillating drop experiments, as ref32, numerous studies37−41 showed that the conservation modulus E′ and the loss modulus E″ of the interface are in agreement with the behavior of a 2D gel near the gelation point43,44

ω ′ ∝ ″ ∝

E E z (1)

where ω is the angular frequency of the oscillation and the

exponent z is given by π = ″ ′ −i k jjj y_{zzz z E E 2 tan 1 (2)

This behavior is generally attributed to the presence at the interface of asphaltene clusters of diﬀerent sizes that result from the aggregation of nanoclusters. The corresponding interface rheology is therefore of an intrinsic nature. Recently, this

mechanism was questioned in ref 45, where a model was

proposed based on a compositional rheology. In this approach, the measured elastic moduli are predicted by an extension of the LVDT approach, which considers two surfactants having

diﬀerent diﬀusion and adsorption dynamics. This alternative

interpretation gave rise to a vigorous debate.46

Regarding low-frequency shear rheology, the conservation

and loss moduli, G′ and G″, of oil−water interfaces with

asphaltenes have been found to follow scaling law 1−2 as

well.34,35However, the given interpretation is diﬀerent from that of a critical-gel behavior. Instead of cross-linking between asphaltenes, a mechanism of jamming is considered, which is described by a two-dimensional version of the soft-glassy rheology model (SGR) developed in refs.47,48This model was ﬁrst applied to interfacial shear rheology to describe

micron-sized spheres,49 Langmuir monolayers of polymers,50 and

carbon nanoparticles51atﬂuid interfaces. The SGR model was then applied to asphaltenes in refs,34,35with a noise temperature

x = z + 1 in the range between 1 and 2, which ensures that G′ and G″ follows relations 1−2. Thus, although dilatational and shear moduli were found to exhibit the same scaling law, the critical-gel approach and the SGR model proposed to interpret them are based on diﬀerent types of physicochemical interactions. It is however worth mentioning that both of them describe an intrinsic rheology.

With regard to high frequencies, we are not aware of any

experimental studies of the rheology of crude oil−water

interfaces. Determination of surface moduli at frequencies up to 100 Hz has been attempted by shifting the rheological curves

obtained at low frequency for various temperatures32 by

assuming a gel-type behavior, or for various surface pressures/ aging times34,51in the framework of the SGR model. However, no direct measurements at high frequencies were achieved so far. The present work reports direct measurements of the shape-oscillation modes of a droplet of crude oil diluted in heptane immersed in water. The oscillating pendant drop method has been adapted to deal with high frequencies.52A droplet of a few millimeters is attached to the tip of a capillary and forced to oscillate at a frequency up to 200 Hz by a periodic injection of a small amount of oil. The drop response is recorded by a high-speed video camera and the time-evolution of its shape is obtained by digital video processing. Eigenfrequencies and

damping rates of modes 2−4 are then determined from the

evolution of their amplitudes as a function of the forcing frequency. Compared to a droplet of pure heptane, both the frequencies and the damping rates are increased. This dynamic characterization is an important result that opens the way to the

prediction of a droplet deformation in a turbulent ﬂow6

involving complex interfaces. Furthermore, it also allows us to discuss the underlying rheology from theories relating surface viscosity and elasticity (2) to eigenfrequencies and damping rate (3). The results exhibit two regimes according to the age of the interface. The rheology of a moderately aged interface is similar at high and low frequencies, whereas that of an aged interface exhibits a diﬀerent behavior. However, in both cases, intrinsic rheology plays a major role.

The article is organized as follows. Weﬁrst present the ﬂuid system under consideration and a characterization of the interface dilatational rheology at low frequency. Then, we describe the experimental determination of shape eigenmodes

and draw conclusions about the inﬂuence of surface-active

agents on the interfacial dynamics. Finally, we discuss the underlying rheology according to the interface age from the theory of eigenmodes.

■

MATERIALS AND LOW-FREQUENCY DILATATIONAL RHEOLOGY

Demineralized (Milli-Q) water is used as the outer phase. The inner phase is a mixture of light crude oil and n-heptane (p.a. grade, Sigma-Aldrich, used as purchased). The crude-oil concentration in the mixture was 10% (w/w). The physical properties of the crude oil used in this study are reported inTable 1, alongside with the saturate, aromatic, resin and asphaltene (SARA) analysis results of the 344+ fraction.

Using diluted crude oil as the drop phase is a deliberate choice. First, compared to undiluted oil, diluted crude oil has a viscosity low enough for the viscous stress not to screen the interfacial dynamics. The viscosity and the density of the diluted oil are substantially the same as those of n-heptane (ηi= 0.4 mPa s,ρi= 680 kg/m3at 25°C), allowing a

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components coexist and interact. The diluted crude oil−water interfacial rheology reﬂects this complexity.

The crude oil−water interface rheology has been characterized by the classic low-frequency oscillating pendant drop method,13using a Tracker dynamic drop tensiometer from Teclis Scientiﬁc. Diﬀerent interface ages have been considered, each of them corresponding to a quasistatic interfacial tensionγe. Small sinusoidal volume oscillations

(δϑ cos(ωt)) are imposed to a droplet that is attached to a capillary tube. At low frequency, the interface shape is controlled by a quasistatic balance between capillary and gravity forces, from which the dynamic interfacial tensionγd(t) can be determined. Denoting A the interface

area, the surface elastic modulus E is obtained from

γ ω ϕ = + A E t d dln cos( ) d (3) The conservation modulus, E′ = E cos(ϕ), is associated to potential forces, whereas the loss modulus, E″ = E sin(ϕ), is related to dissipative ones. As the oscillations correspond to a global inﬂation/deﬂation of the droplet, the interface experiences an almost uniform dilatation. In the case where the interface behavior is due to intrinsic rheology, Ed= E′

should therefore be interpreted as a surface dilation elasticity andηd=

E″/ω as a surface dilatation viscosity.

Experiments have been carried out at 25°C. The amplitude of area variations was set to 5% and oscillation frequencies range between 0.1 and 1 Hz. This ensures linearity between droplet volume and area variations, as well as negligible inertial and viscous eﬀects within bulk ﬂuids.Figure 1a presents the evolution with the interface age of the static interfacial tension γeas well as the conservation and the loss

moduli. The origin of age is taken at the end of the process of droplet formation at the tip of the capillary. As surface-active molecules can adsorb at the interface during droplet formation, the origin of age thus does not correspond to a heptane−water interface. Indeed, the ﬁrst measurement performed at 114 s gives a value ofγeequal to 27.6 mN/

m, which is much lower than that measured with heptane alone (γe= 47

mN/m). Therefore, none of the present data corresponds to a young interface with a small surface coverage, and it is worth noting that in any case the conservation modulus E′ is more than twice larger than the loss modulus E″.

Figure 1displays the existence of two distinct regimes of aging. The ﬁrst regime starts a few hundred seconds after droplet formation until approximatelyﬁve thousand seconds. It is characterized by a very slow decrease ofγefrom 27.6 to 24 mN/m and an increase of both E′ and E″

as the power 0.25 of the interface age, their ratio remaining constant (E″/E′ ≈ 0.41). Beyond 5000 s, a second regime develops where the decrease ofγebecomes steeper whereas the increase of E′ is accelerated

and E″ starts decreasing. From that point, the ratio E″/E′ continuously decreases, becoming smaller than 1/3 for a time larger than 5.5× 104_s.

As many previous studies32,34,35,37−41have found that elastic moduli of oil−water interfaces containing asphaltenes did follow scaling law 1− 2, this law has been evaluated on the present data.Figure 1b shows the evolution with age of the exponent z computed from the experimental values of E″/E′ byeq 2. During thefirst regime, z is found constant and equal to 0.25, whereas during the second regime (≳5000 s) it decreases exponentially with age.Figure 2shows the evolution of E′ and E″ with frequency for two different ages within the first regime. In agreement

witheq 1, both moduli increase asωzwith the value z = 0.25 given byeq 2.

This good agreement with scaling law 1−2 cannot, however, lead us to conclude about the underlying physicochemical mechanism. Let us discuss the interpretation in terms of critical-gel or SGR models by considering the features of the aging process. Describing the interface as a 2D-gel near its gelation point was proposed by ref32and considered in subsequent studies37−41 _{for the interpretation of dilatational}

interface elastic moduli of crude oil−water interfaces that were aged enough, typically for more than 14 hours, such as to be in thermodynamic equilibrium with the bulk phases. In the present study, as in ref39, scaling law 1−2 is observed for moderately aged

Table 1. SARA Analysis and Physical Properties of the Crude Oil

SARA analysis of the 344+ fraction

saturates (% w/w) 47.5

aromatics (% w/w) 39.1

resins 12.6

asphaltenes 0.1

mass yield of the fraction (% w/w) 50.76 physical properties of the crude oil

density at 15°C (kg/m3₎ ₈₅₀

viscosity at 20°C (Pa s) 9× 10−3

Figure 1.Evolution of interface properties with age measured by the oscillating drop method at a frequencyω/2π = 0.2 Hz. (a) Static surface tension γe, conservation modulus E′, and loss modulus E″. (b)

Estimation of the exponent of the evolution of E′ and E″ as a function of the frequencyω fromeq 2.

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droplets for which the aging process is not achieved yet, asγeis still

slightly decreasing and especially both E′ and E″ are significantly increasing. It is however not clear for us how the aging process, either by a continuous adsorption of asphaltenes at the interface or by an unceased structuration of asphaltenes clusters, is consistent with the assumption of a 2D critical gel. The compatibility of the observed aging process with the SGR model is even more questionable. Assuming a constant composition of the interface, the SGR model only predicts47 an evolution of the rheology for a noise temperature x lower than unity and, in this case, the loss modulus should decrease whereas the conservation modulus should remain constant. On the other hand, considering a continuous arrival of asphaltenes should increase the crowding and cause a decrease of x. However, during thefirst regime of aging, the exponent z characterizing the ratio E″/E′ remains constant and corresponds to a noise temperature greater than unity x = z + 1 = 1.25. The SGR model seems therefore not consistent with thefirst regime of aging, whereas it is not in direct contradiction with the second regime where z tends toward zero.

At this stage, it is diﬃcult to conclude about the nature of the aging process, which can be associated with either an evolution of the surface coverage governed by diﬀusion or a reorganization of the adsorbed molecules on the interface, or with both of them. Furthermore, the mechanisms causing the interface elastic moduli to follow scaling law 1−2, especially for a system which is still experiencing such an aging process, are still unknown. However, this scaling law seems to be a characteristic feature of the low-frequency rheology of crude oil−water interfaces. An important question is thus whether these dependences of the interfacial properties with the frequency is still valid in free droplet oscillations, which involve high frequencies and interface deformation implying both area dilation and shear. The next section will present investigations of the dynamics of a diluted crude-oil droplet oscillating at a high frequency in water.

■

HIGH-FREQUENCY DYNAMICS OF AN OSCILLATING DROPLET

Experimental Method. A pendant drop dynamic tensi-ometer is used to generate high-frequency droplet oscillations. It

is a DSA 100 from Krüss equipped with an oscillating drop

module (ODM) able to generate oscillations up to 200 Hz.

Figure 3shows a schematic of the setup. A syringe (1) is used to form an oil droplet at the end of a capillary immersed in water

(4). A piezoelectric membrane (2″) mounted on a wall of the

oil-storing cell (2) generates small droplet-volume oscillations at

a given frequency ω. The droplet is illuminated by a

light-emitting diode (LED) backlight panel (3) and recorded by a Photron APX high-speed camera (5) operated at a frame rate

ranging between 60 and 2000 Hz, depending onω. Then, the

droplet shape is obtained at each instant by digital video processing of the images. Full details on the experimental technique are given in ref52where the method has been tested and validated for heptane droplets in water. Two conclusions of this previous work deserve to be mentioned. First, the limits of the linear regime of oscillations have been determined. Second, the roles of both gravity and attachment of the droplet to the capillary tip upon the eigenmodes of shape-oscillations have been shown to be negligible, provided the droplet size is large enough. The linear theory of a droplet oscillating around a spherical shape in the absence of gravity is thus relevant to interpret the results of the present study.

Small droplet-volume oscillations drive interface-shape oscillations. In order to be analyzed, shape oscillations need to be decomposed into spherical harmonics, which correspond to the eigenmodes of oscillations. As the droplet remains axisymmetric, its shape at each instant is fully characterized by the local radial position r of the interface as a function of polar

angleθ (seeFigure 3). The local radius of the interface is thus written as

∑

θ = + = i k jjjjj j y { zzzzz z r t( ) a A t( ) A t P( ) (cos( )) n N n n 0 2 (4)

where a is the radius of the sphere of same volumeϑ = 4/3πa3as that of the droplet, A₀(t) stands for the volume variation, and An(t) is the instantaneous dimensionless amplitude of spherical

harmonic n, which is deﬁned by Legendre polynomial P_n.

Rigorously, the summation should expand to inﬁnity. However,

it turns out that it is suﬃcient to stop to harmonic 10 to

accurately describe the experimental shapes. At each instant, the A_nare determined byﬁtting the experimental droplet contours usingeq 4with N = 10. As the equilibrium shape of the droplet is not spherical, the time average is subtracted from each A_n(t) signal, so that only theﬂuctuating part ΔAn(t) is kept. In the

following, we present results concerning the threeﬁrst modes of shape oscillations, which correspond to n = 2, 3, and 4.Figure 4

illustrates the time evolution ofΔA_n(t). As oscillations lie in the linear regime, we observe perfect sinusoidal functionsΔAn(t) =

δAn cos(ωt + ϕ), from which we can determine the peak

amplitude δAn within an accuracy of 5%. This procedure is

repeated for various forcing frequenciesω in order to obtain a resonance curveδAn(ω) for each mode. Owing to the design of

the ODM, the amplitude of the oscillating forcing volumeδϑ

cannot be maintained constant asω is varied. Therefore, each δAn(ω) needs to be divided by the relative amplitude of volume

variations: Ãn(ω) = δAn(ω)/(δϑ(ω)/ϑ).

Various Types of Interfaces Considered. Three types of interfaces are considered, which diﬀer by their state of aging (see

Table 2). Type T0 is the reference case of a droplet of heptane. Type T1 corresponds to a diluted crude-oil droplet of age 1200 s, which therefore belongs to theﬁrst aging regime displayed in

Figure 1. Considering older interfaces belonging to the second

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regime of aging was more problematic. Because of a lack of a regulation system of the volume, the DAS 100 tensiometer did not allow one to maintain a pendant droplet for 5000 s or more. To overcome this limitation, type T2 has been obtained by provoking an artiﬁcial aging of the interface (pictures 6a−c in

Figure 3), based on the idea that reducing the interface area increases the concentration of adsorbed molecules in a similar way as aging. A droplet is formed at the capillary tip and let in contact with water during initial aging time Δt1. Then, the

droplet is shrunk by sucking a given amount of oil until wrinkles

appear (picture 7 in Figure 3). Then, the droplet volume is

slightly increased back, just enough for the wrinkles to disappear. Finally, the droplet is let to relax for a duration Δt2 before

starting oscillations. Moderate initial aging time and relaxation

time are applied to type T2A (Δt1 = 1200 s, Δt2 = 300 s),

whereas longer ones are applied to type T2B (Δt1= 3600 s,Δt2=

1200 s). It is important to stress that, if reducing the interface area is very likely to increase surface coverage, we ignore if it causes a reorganization of surfactants at the interface which is similar to aging. For that reason, we will not claim that interfaces T2 really correspond to the second regime of aging identiﬁed in

Figure 1. Type T2 will be used to probe whether results obtained with type T1 still hold for interfaces having experienced a more complex history. In the following, type T2 will be referred as to “artiﬁcially aged interfaces”.

Resonance Curves of Shape Oscillations.Figures 5−8

display the response in amplitude of modes 2−4 as a function of the forcing frequencyω for the various types of interfaces under consideration. Symbols correspond to experimental data.

Continuous lines show the theoretical resonance curve Ãn(ω)

of a linear harmonic oscillator

ω β ω ω ω β ω ̃ = ̃ − + A ( ) 2 A ( ) 4 n n n n n n

exp exp max exp2 2 exp2 2

(5) Figure 4.Time evolution of the amplitudes of spherical harmonics

characterizing interface-shape oscillation (example for T1 interface).

Table 2. Various Types of Interfaces Considered in High-Frequency Experiments

interface type T0 T1 T2A T2B

innerﬂuid heptane diluted crude diluted crude diluted crude

oil + heptane oil + heptane oil + heptane

outerﬂuid water water water water

droplet diameter (mm) 4.48 3.73 3.06 3.27

aging timeΔt1(s) 1200 1200 3600

contraction/expansion cycle no no yes yes

relaxation time after expansionΔt2(s) 300 1200

static interfacial tensionγe(mN/m) 47 25 20 20

Figure 5.Amplitudes Ãnof interface modes 2−4 as a function of the

forcing frequency for a drop of heptane in water (type T0). Symbols: measurements. Lines: bestﬁt of each peak by a harmonic oscillator (eq 5).

forcing frequency for a moderately aged diluted crude-oil droplet in water (type T1). Symbols: experimental data. Continuous lines: bestﬁt of each peak by a harmonic oscillator (eq 5). Dashed lines: prediction from theory by using values of interface rheological properties extrapolated from low-frequency characterization: Es = 2Ed =

2E′extra_{(ω) and η}

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where the angular frequencyωnexp, the damping rateβnexp, and the

maximum at resonance Ã_nmax_{have been adjusted to provide the}

bestﬁt of the peak of each mode n. Experimental frequencies and damping rates are reported inTables 3−6, together with the

theoretical valuesωnrefandβnrefobtained by assuming an interface

described by a constant interfacial tension.3 The considered values ofγeare given inTable 2and correspond to the static

interfacial tension measured for each interface type. These reference frequencies and damping rates thus only account for the presence of surface-active agents through the values of the interfacial tension but ignore the complex rheology of the interface.

The case of the heptane droplet (T0) is used to determine to which extent the theory of a free droplet in the absence of gravity is relevant to interpret the present experiments. As expected, it is in fairly good agreement with the theory of a pure interface. The resonance curve of each mode (Figure 5) shows a narrow peak. The experimental values of the frequencies are within 2% of the theoretical values for mode 2 and 4% for mode 4, whereas the

discrepancy on the damping rate is about 20% (Table 3). As

discussed in ref52, the larger discrepancy inβncan be attributed

to the eﬀect of the attachment of the droplet to the capillary. In the following, these discrepancies will be considered as reasonable estimates of overall uncertainties made in the determination ofωnandβnfor interfaces of types T1 and T2.

Let us now consider a moderately aged interface (T1). The

resonance curves (Figure 6) are shifted toward higher

frequencies whereas the peaks are much broader. Table 4

shows that experimental frequencies are larger than reference ones by 57% for mode 2, 25% for mode 3, and 8.1% for mode 4. Moreover, experimental damping rates are larger than reference ones: six times larger for mode 2, four times for mode 3, and three times for mode 4.

The effect of surface-active agents is even more stronger for artificially aged interfaces. Interfaces T2A (Figure 7) and T2B (Figure 8) show very similar resonance curves. The peaks strongly moved toward high frequencies and are very much wider so that they partly overlap. For mode 2, the experimental eigenfrequencies are about three times larger than the reference frequency, whereas the experimental damping rates are more than 12 times larger than the reference value (Tables 5and6). We can therefore conclude that surface-active agents that are present in crude oil significantly change the mechanical behavior of the interface at frequencies around 100 Hz. As well-defined resonance peaks are detected for crude oil−water interfaces of any age, the present method is able to characterize the dynamics of shape oscillations of droplets having a complex interface. The eigenfrequencies and damping rates of shape eigenmodes can therefore be directly measured. Moreover, as eigenfrequencies

forcing frequency for an artiﬁcially aged crude-oil droplet in water (type T2A). Symbols: measurements. Continuous lines: bestﬁt of each peak by a harmonic oscillator (eq 5).

Figure 8. Amplitudes of interface modes 2−4 as a function of the forcing frequency for an artiﬁcially aged crude-oil droplet in water (type T2B). Symbols: measurements. Continuous lines: bestﬁt of each peak by a harmonic oscillator (eq 5).

Table 3. Frequencies and Damping Rates of Eigenmodes of a 4.48 mm Heptane Drop in Water (Type T0)

mode ωexp_(rad/s) _ωref_(rad/s) _βexp_(s−1₎ _βref_(s−1₎

2 151 154 5.85 4.67

3 279 289 11.61 8.82

4 423 440 15.81 13.90

Table 4. Frequencies and Damping Rates of Eigenmodes for a Moderately Aged Diluted Crude-Oil Droplet in Water (Type T1)

2 230 146 34 5.62

3 345 275 48 10.62

4 453 419 52 16.75

Table 5. Frequencies and Damping Rates of Eigenmodes for

an Artiﬁcially Aged Diluted Crude-Oil Droplet in Water

(T2A)

2 490 175 110 7.60

3 580 328 95 14.35

4 735 500 95 22.64

Table 6. Frequencies and Damping Rates of Eigenmodes for

an Artiﬁcially Aged Diluted Crude-Oil Droplet in Water

(T2B)

2 540 158 86 6.76

3 628 297 84 12.75

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are in general more sensitive to surface elasticity, whereas

damping rates are more inﬂuenced by surface viscosity, the

present results suggest that the diluted crude oil−water interface features signiﬁcant viscoelastic properties, which depend on the aging process. In the next section, we attempt to infer the surface elasticities and viscosities from the resonance curves and discuss the relation between low- and high-frequency interfacial rheology.

■

DISCUSSION ON INTERFACIAL RHEOLOGY

Theoretical Background. Our objective is to determine the elasticity and the viscosity of the interface from the measured eigenfrequencies and damping rates. This can be done by solving the coupled dynamic equations for theﬂuids and the interface. However, it requiresﬁrst to assume a model that describes the interfacial rheology. Let us start by reviewing the various possible origins of the viscoelastic properties of the interface.

The ﬁrst cause of complex interfacial rheology is of a

compositional nature and only concerns dilatation. It arises from

the fact that surface energy γ depends upon the surface

concentration of surface-active agents and gives birth to Gibbs elasticity. For a single surfactant at surface concentrationΓ, it writes: E_Gibbs=−dγ/dln Γ. (For multiple surfactants we have to consider the sum of all contributions to the Gibbs elasticity.) For an insoluble surfactant, the number of molecules adsorbed on the surface is constant and the variations of concentration are simply related to the interface area A by dlnΓ = −dln A. In this

case, the conservation modulus E′ measured in low-frequency

oscillating pendant drop experiments is thus equal to E_Gibbs. When molecule transfer occurs between the interface and the droplet bulk, E′ is in general lower than E_Gibbs. It decreases when transfer becomes faster as the magnitude of concentration variations during an oscillating period is attenuated. The time scale of transport of surfactant molecules through the bulk is controlled by diﬀusion in the vicinity of the interface and writes:

τ =

( )

_∂∂Γ

D C

diff

1 2

, where C and D are respectively the surfactant concentration and diffusion coefficient in the droplet fluid. The dimensionless group λ = (ωτdiff)−1/2 is then introduced11 to

compare the time scale of diﬀusion to the oscillating period, the caseλ = 0 corresponding to negligible molecule transfer. The eﬀective compositional elasticity is thus controlled by both E_Gibbs andλ. As molecule transfers are associated with a dissipative

motion in the ﬂuid, a loss modulus is also measured in

low-frequency oscillating pendant drop experiments when λ ≠ 0,

even in the absence of any real interface viscosity. The same mechanism can be responsible for an increase of the damping rate of droplet shape eigenmodes at a high frequency.

The second cause of interfacial viscoelasticity is related to interactions between adsorbed molecules, which give rise to intrinsic surface elasticity and viscosity. The interface should thus be considered as a two-dimensional viscoelastic material, which is characterized by four physical parameters: a dilatational elasticity E_d, a shear elasticity E_s, a dilatational viscosityη_d, and a shear viscosityηs.

Assuming small amplitude oscillations, the dynamic equations describing the droplet oscillations can be linearized. The eigenfrequency and the damping rate of each mode are then determined by cancelling the determinant of a matrix, which leads to solve a transcendental equation. This equation wasﬁrst derived in two seminal works. First, ref3considered the case of an elastic interface involving interfacial tension (γ_e), dilatational and shear elasticities (Ed, Es), as well as dilatational and shear

viscosities (η_d,η_s). Then, ref12addressed the case of viscoelastic

interface involving compositional rheology (EGibbs, λ) and

intrinsic viscosity (η_d, η_s). By completing the determinant of ref12by the terms corresponding to intrinsic elasticity derived

by ref 3, the general equation for an interface including

compositional rheology as well as both intrinsic elasticity and

viscosity is obtained (see Appendix I for details). In what

follows, this equation is solved numerically to determine the values ofω_nandβ_n.

Moderately Aged Interface (T1). We start by analyzing moderately aged interface T1, which belongs to thefirst regime of aging identified from low-frequency dilatational rheology. In this regime, interfaces are still aging but yet show a significant viscoelastic rheology with E′ > E″. For oscillating frequencies ω in the range from 0.63 to 6.3 rad/s, both moduli have been found to evolve asω0.25. In order to assess whether this behavior still

holds at high frequency, the functions E′(ω) and E″(ω)

determined at low frequency (Figure 2) have been extrapolated to higher frequencies by assuming that theω0.25_{law is still valid.}

Extrapolated values evaluated at the peak frequency of the resonance curves, E′extra₍_ω

n

exp_{) and E}_″extra₍_ω n

exp_{), are reported in}

Table 7, where E″extrahas been divided by ωnexp to allow its

interpretation as a surface viscosity. Values of E′extra₍_ω n exp_{) are}

found to be of the order of 0.05 N/m whereas those of E″extra₍_ω

n exp_)/_ω

n

exp_{are of the order of 10}−4_N_·s/m.

The model of the interface described above involves six a priori unknown parameters (E_Gibbs,λ, E_d, E_s,η_d, η_s). It is too much to attempt a brute-force determination of the parameters

by searching for the values that provide the best ﬁt of the

experimental values ofωnandβn. We thus adopt a step-by-step

approach.

Considering only static interfacial tension, the eigenfrequency of mode 2 should beω₂ref_{= 146.5 rad/s, whereas the measured}

value isω2exp = 230 rad/s. Such an important increase in the

frequency is expected to be the signature of a surface elasticity.

Let us ﬁrst assess whether it can be due to a compositional

elasticity.Figure 9shows the theoretical value ofω₂as a function of EGibbsfor various values ofλ. It turns out that ω2is not a

monotonic function of E_Gibbsand its value never exceedsω₂ref_by

more than 3%. In the absence of molecule transfer between the bulk and the interface (λ = 0), ω₂ﬁrst increases from the initial value 146.5 rad/s up to reach a maximum at 149.8 rad/s for E_Gibbs= 1.2× 10−3N/m; then it continuously decreases as E_Gibbs further increases. Accounting for molecule transfer (λ > 0) simply moves the maximum to the right and reduces its magnitude. It is worth mentioning that, in the absence of molecule transfer, the eﬀect of the Gibbs elasticity E_Gibbs is exactly the same as that of an intrinsic dilatational elasticity Ed. In

other words, the functionω₂(E_Gibbs) forλ = 0 is the same as the functionω2(Ed), which does not depend onλ. This leads to an

important conclusion. The strong increase inω₂cannot be the result of only a compositional rheology. More generally, it can Table 7. Conservative and Loss Interface Moduli

Extrapolated from Low-Frequency Dilatational Rheology

Measurements for a Moderately Aged Diluted Crude Oil−

Water Interface (T1)

mode ωexp_(rad/s) _E_′extra_(N/m) _E_″extra_/_ωexp_(N_·s/m)

2 230 0.053 9.5× 10−5

3 345 0.059 7.0× 10−5

(11)

neither be explained by only considering any kind of dilatational elasticity. Because we are dealing with high frequencies, we will setλ = 0 in what follows. This implies that there will be no more

distinction between E_Gibbs and E_d regarding the comparison

between the prediction of the model and experimental resonant curves.

Figure 10showsω2as a function of the shear elasticity Esfor

various dilatational elasticities E_d. The functionω₂(E_s) for E_d= 0

isﬂat. Taken together, results of Figures 9and 10show that increasing either the dilatational or the shear elasticity while the other one is set to zero does not lead to a signiﬁcant increase of ω2. However, when neither of the moduli is negligible,ω2turns

out to be an increasing function of both Edand Es. Moreover,

when E_dand E_sare of the same order of magnitude, the value ω2exp= 230 rad/s can be reached for an elasticity of the order of

the extrapolated conservation modulus E′extra₍_ω 2

exp_{) = 0.053 N/}

m.

Figure 11shows the theoretical damping rateβ₂of mode 2 as a

function of the dilatational viscosity ηd for various shear

viscositiesη_s and E_s = E_d = 0.053 N/m. The value of β₂are increasing functions of both ηd and ηs. Similar to what was

observed for the eﬀect of elasticity upon frequency, it is possible to retrieve the experimental damping rate,β2exp= 34 s−1, with

values of both viscosities that are close to the extrapolated value E″extra₍_ω

2 exp_)/_ω

2

exp_{= 9.5}_{× 10}−5_N_·s/m.

Assuming a viscoelastic interface with both (1) an elastic dilatational elasticity and a shear elasticity of the same order and (2) a dilatational viscosity and a shear viscosity of the same order is therefore compatible with the observed shape-oscillation dynamics. Moreover, the measured eigenfrequency and damp-ing rate of mode 2 can be recovered with values of elasticities and viscosities close to the values obtained by extrapolating the conservation and loss modulus measured at a low frequency by assuming that theω0.25 law is still valid at a high frequency. Unfortunately, there is no unique set of parameters (Ed, Es,ηd,

ηs) that leads to the experimental values ω2exp and β2exp. For

example, Figure 10 shows that ω2exp can be reached for two

diﬀerent values of Esdepending on whether Edis assumed to be

equal to Es or twice as large. If we consider that the functions

E′(ω2) and E″(ω) obtained from low-frequency dilatational

rheology are valid in the whole range of frequency investigated in this work, the dilatational elasticity and viscosity should be given by

ω = ′ ω ∝ ω

E_d( ) Eextra( ) 0.25 (6)

η ω_d( )= ″E extra( )/ω ω∝ω−0.75 ₍₇₎

However, if the ratios Es/Edandηs/ηdare known to be of

order 1, their exact values remain undetermined. To go further, the resonance curves Ãn(ω) of modes 2−4 have been computed

by the following method. First of all, the ratios E_s/E_dandη_s/η_d are set to a given value. For a given forcing frequencyω, the values of E_dandη_dare determined byeqs 6and7, whereas those of Es andηs are given by the prescribed ratios. The dynamic

parameters ω_n and β_n are computed by solving the

tran-scendental equation which describes the dynamics for that given set of surface viscosity and elasticity. Then, the mode amplitude Ãnis calculated by using eq 5. Finally, the resonance curves

Ã_n(ω) are obtained by doing the same calculation for all ω. The best results, obtained for Es/Ed= 2 andηs/ηd= 2, are

reported in Figure 6 (dashed lines). (Other examples of

theoretical resonant curves are shown inFigure 14ofAppendix II.) Regarding that the only adjustable parameters are the ratios Es/Edandηd/ηd, the agreement with the experimental curves is

remarkable. Only a slight discrepancy on the peak frequency of mode 4 is to be noted. Except from that, the peak frequency and

Figure 9.Mode-2 frequency as a function of the Gibbs (or intrinsic dilatational) surface elasticity (physical parameters of case T1 and Es=

ηd=ηs= 0).

Figure 10.Mode-2 frequency as a function of the shear surface elasticity for various dilatational surface elasticities (physical parameters of case T1 and EGibbs=λ = ηd=ηs= 0).

(12)

the peak width of all three modes are very well reproduced. We can therefore conclude that the evolutions of the surface elasticity asω0.25and of the surface viscosity asω−0.75are valid for the frequency range of the three modes.

Regarding moderately aged interface T1, the results thus indicate that the rheology of the crude oil−water interface, which controls droplet-shape oscillations in the range of

frequencies between 10 and 80 Hz, is not diﬀerent from that

governing quasistatic oscillations at frequencies smaller than 1 Hz. The interface behaves as a 2D viscoelastic material with shear elasticity and viscosity that are of the same order of magnitude as their dilatational counterparts and are in agreement with scaling law 1−2.

The fact that the bestfit of the experimental resonant curves is obtained for E_s > E_ddeserves discussion. If we interpret the observed rheological behavior as the existence of real 2D membrane, its material would be auxetic, which would be hardly believable and in conflict with our assumption of 2D isotropy. However, even if we consider that there exists a membrane of constant composition, its structure is three-dimensional at the molecular level and there is no reason to extend the isotropy property of the interface plane in the normal direction. When the in-plane area increases, its thickness may decrease because of rearrangement or rotation of asphaltenes clusters: we can therefore conclude nothing about the volume variation of this material. That being said, it is also possible that the composition of the interface changes as its area varies. When oscillations are rapid (λ = 0), the number of surface-active molecules at the interface remains constant but more or less water or oil molecules can occupy the spaces between them. Thus, because either the organization of the surfactant in the direction normal to the interface or composition may change during the oscillations, the interface is not a real 2D material and there is, in principle, no reason to reject a description involving an effective 2D rheology with Es> Ed. However, even if the bestfit is

obtained with E_s> E_d, the discrepancy with E_s= E_dis not so large. It is therefore reasonable to limit our conclusion to the fact that E_sand E_dare of the same order.

Once an eﬀective rheology has been inferred from the

dynamics, the underlying physicochemical mechanisms can be discussed. Regarding shear elasticity and viscosity, there is no doubt that they are of an intrinsic nature and imply interactions between adsorbed molecules or molecules clusters. However, it

is diﬃcult to conclude whether these interactions imply

molecular bounds, like in a gel, or steric eﬀects, like in a soft glassy material. Regarding dilatational rheology, the question of its compositional or intrinsic nature is more complex and requires to distinguish between elasticity and viscosity. As already mentioned, the results of the present model do not make any diﬀerence between an intrinsic dilatational elasticity and a compositional Gibbs elasticity. There is no reason to think that Gibbs elasticity is null, especially at a high frequency for which variations of surface concentration are expected to be directly related to area variations. The observed value of Edcould thus a

priori be due to any combination of compositional and intrinsic contributions. Furthermore, as we have considered that oscillations are rapid compared to diffusion time (λ = 0) and thus assumed insoluble surfactant, all damping effects modeled by intrinsic viscositiesηsandηddo not include any effect because

of changes in interface composition. However, if there existed a dissipation related to an irreversible process associated to periodic changes in interface composition, this eﬀect would be

imbedded in the observed value of ηd. Because values of

dilatational elasticity and viscosity are found to be strongly correlated to their shear counterparts, we are prone to think that dilatational rheology is predominantly of an intrinsic nature. Having no certainty about the physicochemical structure of the interface, we cannot preclude a coupling between compositional and intrinsic mechanisms that would lead to correlate the values of elasticity and viscosity.

Artiﬁcially Aged Interface (T2). As indicated by the large peak frequencies of the resonance curves plotted inFigures 7

and8, T2 interfaces exhibit a large elasticity. According to the way they are produced, it is reasonable to think that their coverage rate is close to the interface saturation. We a priori

expect them to be representative of ages far beyond theﬁrst

regime of aging, which ends at 5000 s. However, because of their complex history a direct comparison between low- and high-frequency rheological characterization is therefore not possible for them. As a consequence, we have no clue to estimate either the values of elasticity and viscosity at high frequency or their possible dependence on frequency. Therefore, we cannot use the same method as for T1 interfaces where the resonance curves have been computed by using expected values of Edand

ηd. We shall thus attempt to infer rheological properties from

measured values ofω_nexp_and_β n exp_.

On the basis of what we learnt from the T1 interface, we attempt to describe T2 interfaces by a two-dimensional viscoelastic rheology. Here, we present results obtained by keeping the aspect ratios that gave the bestﬁt for interface T1: E_s/E_d = 2 and η_s/η_d = 2. Note that similar conclusions are obtained for other ratios of order 1. (In particular, the results obtained for Es/Ed= 1 andηs/ηd= 1 are provided inFigures 15

and 16 of Appendix II.) The two remaining free rheological parameters, Edandηd, are then adjusted so that the values ofωn

andβnobtained by solving the dynamic equations match the

experimental valuesω_nexp_and_β n

exp_{reported in}_{Tables 5}_and ₆_.

The determination of E_dandη_dis carried out independently for

each mode of interfaces T2A and T2B. Figure 12shows the

moduli E′ = Edand E″ = ωnηdobtained by this method as a

function of the frequency. As three eigenmodes have been investigated, three measurements points are available for each modulus and each interface type. The moduli E′ and E″ of mode 2 of T2A and T2B interfaces (ﬁrst points from the left inFigure

Figure 12.Conservation (E′ = Ed) and loss (E″ = ωnηd) moduli

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12) are found to be 5−10 times larger than those of T1 interface. However, the range of frequencies available for T2 interface spreads from 490 rad/s (ω₂exp _{of T2A) to 795 rad/s (}_ω

4 exp_of

T2B), above the range of the T1 interface (230≤ ωnexp≤ 453

rad/s). We must keep in mind that the investigation of T2 interfaces thus diﬀers from that of the T1 interface by both the aging and the frequency range.

The conservation modulus of T2B interface, which is older, is about twice as large as that of the T2A interface. However, the evolution of elasticity with frequency is similar for the two types of interfaces. We observe a signiﬁcant decrease between 500 and 600 rad/s and then a plateau between 600 and 800 rad/s. On the other hand, the loss moduli of T2A and T2B interfaces have similar values and do not display any signiﬁcant evolution with ωn. These trends are clearly not in agreement with theω0.25

increase of both moduli observed for the T1 interface. The

disparity of behavior with the T1 interface is conﬁrmed by

Figure 13, which shows the modulus ratio E″/E′ as a function of

frequency. Regarding T2A, the ratio starts from a value (E″/E′ = 0.42 atωn= 490 rad/s) close to that of T1 (E″/E′ = 0.41), but it

then decreases strongly to reach E″/E′ = 0.32 at ωn= 735 rad/s.

Regarding T2B, E″/E′ is approximately constant and equal to

0.24 throughout the available frequency range (540≤ ω ≤ 795

rad/s).

The major discrepancy between the interfacial rheology of T1 and T2 interfaces lies in the diﬀerent evolution of the moduli with the frequency. T2 interfaces are not compatible with scaling law 1−2 for frequencies larger than 500 rad/s. Results obtained with the T2A interface show that this might result from only the diﬀerence in the considered range of frequency, as the results are

compatible with theω0.25 power law and a constant modulus

ratio (E″/E′ ≈ 0.4) for ω below 500 rad/s and a decrease of

elasticity beyond. This interpretation cannot be conﬁrmed by high-frequency T2B-interface results, which start at 540 rad/s, where the modulus ratio is already as small as 0.26. We can thus

not deﬁnitively conclude whether the diﬀerences in the

viscoelastic properties of T1 and T2 should be ascribed to either the frequency range or the aging process, or to both of them.

■

CONCLUSIONS

The dynamic response to periodic forced oscillations of a diluted crude-oil droplet in water has been investigated at a low

frequency by classical dilatation experiments and, for theﬁrst time, at a high frequency by exciting inertial modes of shape oscillations.

For frequenciesω/2π lower than 1 Hz, the droplet shape is

controlled by a simple quasistatic balance between buoyancy and surface tension, providing a direct characterization of the dilatational rheology of the interface by the determination of the

conservation modulus E′ and the loss modulus E″. Because of

the process of formation, the coverage rate of the droplet by

surfactants is signiﬁcant and its interface shows from the

beginning a substantial viscoelastic behavior with E′ > E″. However, the interface is not at equilibrium and an aging process is observed, which displays two successive stages. Aﬁrst regime is observed for aging times between a few hundred and 5000 s, where both moduli increase with time but their ratio remains constant. In addition, E′ and E″ both increase with the forcing

frequency asωz, with = _π − ″ = ′

( )

z tan E 0.25 E 2 1 . Such a scaling law has been reported in many previous works involving crude oil either for surface dilatational moduli32,37−41and interpreted as a 2D critical gel, or for shear moduli34,35and interpreted as a 2D soft glass material in the framework of the SGR model. In the present case, because of the aging process that is taking place, it is not clear whether any of these interpretations can apply. However, this scaling law seems to be a characteristic of crude oil

or model ﬂuids involving asphaltenes and resins. A second

regime of aging occurs at a larger time where the conservation modulus increases faster with age, whereas the loss modulus decreases, indicating that surface elasticity becomes more and more predominant.

For frequencies between 10 and 200 Hz, inertial modes of droplet shape oscillations are excited. The interface thus undergoes oscillations which result from the interplay between inertial stresses within theﬂuids and dynamic stresses within the interface, and are damped by viscous stresses. During these oscillations, the interface experiences both nonuniform dilatation and shear. By projecting the instant local radius of the interface into the basis of spherical harmonics, the amplitudes of the shape eigenmodes are determined and resonance curves are obtained, allowing the identiﬁcation of the eigenfrequencyωnand damping rateβnof modes of order n

from 2 to 4. The knowledge of ω_n and β_n is necessary and

sufficient to predict the deformation of a droplet in an unsteady flow. This direct characterization of the interface dynamics makes possible, forfluid systems as complex as crude oil, the prediction of the breakup of a droplet immersed in a turbulent flow, without the need for the exact chemical composition of the fluid system or a description of the interface rheology.

Solving the coupled dynamic equations for thefluids and the interface in the linear regime of oscillation leads to the following conclusions regarding the interface rheology at a high frequency. Two types of interfaces have been considered. Type T1 corresponds to moderately aged interfaces that belong to the first aging regime. Type T2 consists of artificially aged interfaces with a higher coverage rate, which are obtained from a T1 droplet that has been shrunk until the interface displays wrinkles and slightly re-expanded so as wrinkles disappear. In all cases, it is found that the interface rheology is well described by an equivalent two-dimensional isotropic viscoelastic material characterized by dilatational (Ed) and shear (Es) elasticities,

and dilatational (ηd) and shear (ηs) viscosities, which are

functions of the frequencyω. A ﬁrst major result is that the experimental resonance frequencies, which are very large

(14)

compared to those of a surfactant-free interface, can only be reproduced by considering shear and dilatation properties of the same order of magnitude: E_s(ω) ≈ E_d(ω), η_s(ω) ≈ η_s(ω). As E_s andηsare necessarily of an intrinsic nature, we can conclude that

intrinsic rheology plays a major role for such interfaces. The question of the contribution of compositional rheology to Edand

ηdremains an open question. However, the fact that Edandηd

closely behave as Esandηsleaves only two options. Either the

compositional contribution to Edandηdis negligible compared

to that of the intrinsic one, or there exists a strong coupling between compositional and intrinsic mechanisms that explains the similitude between E_dand E_son one side andη_dandη_son the other side. For a moderately aged interface T1, a second major result is found. The resonant curves obtained at a high frequency are well reproduced by extrapolating the low-frequency power law at higher frequencies. Surprisingly, this indicates that the interface rheology does not change between 0.1 and 80 Hz and therefore that low-frequency tensiometry is relevant throughout this range of frequency. For interfaces T2 that have been artificially aged, the elasticity is significantly larger, which causes a lower value of the ratio between the loss and the conservation moduli. In addition, elasticity is observed to decrease with the frequency in between 80 and 100 Hz, and keeps a constant value between 100 and 130 Hz. This behavior is no longer in agreement with the scaling law observed for a moderately aged interface in the range 0.1−80 Hz. This difference can either be due to a transition of rheological properties at larger frequencies or a change in the structure of the network of adsorbed species at a higher coverage rate. In order to answer this question, future work should explore the dynamics of the T1 interface at higher frequencies and of T2 at lower frequencies. The resonant frequencies of the shape eigenmodes depending on the droplet size, this could be done by changing the droplet volume. However, because of technical limitation of the present setup, this will require building a new apparatus.

In summary, the rheology of a crude oil−water interface over a large range of frequencies is well described by considering an equivalent two-dimensional viscoelastic material, the elasticities and viscosities of which vary as a weak power law of the frequency. This does not imply that there really exists a 2D membrane of constant structure and composition at the interface. First, it means that such a model can be used to predict the deformation of a droplet of diluted crude oil in water. Second, it provides serious constraints on the modeling of

physicochemical underlying mechanismsas the irrelevance of

considering a pure compositional interpretationand hopefully gives some hints for further understanding of the structure of crude oil−water interfaces.

■

APPENDIX I

Transcendental Equation for Eigenmode Determination

Problem Statement. This appendix presents the tran-scendental equation that is solved to compute the frequencies and the damping rates of the eigenmodes of oscillations of a droplet with a complex interface. This derivation is closely inspired from the seminal works of refs,3,12to which the reader is referred for further details.

We consider a spherical droplet of radius a at equilibrium. The

inner and the outerﬂuids are Newtonian. The density and the

viscosity of the inner ﬂuid are respectively ρi andηi, whereas

those of the outerﬂuid are ρ_oandη_o. The interface is modeled as a two-dimensional medium characterized by an equilibrium

interfacial tension γ_e, a Gibbs elasticity E_Gibbs, and intrinsic viscoelastic properties: dilatational elasticity Ed, shear elasticity

E_s, dilatational surface viscosityη_d, and shear surface viscosityη_s. We consider a surfactant that is soluble in the outer phase and can be adsorbed on the interface. The diffusivity of the surfactant in the bulkfluid is D, whereas its diffusivity on the interface is DS.

Finally, the surface concentrationΓ of the surfactant on the interface is considered to be at equilibrium with its volume

concentration C in the ﬂuid at the interface. This

thermody-namic equilibrium is characterized by the length scale

Λ = ∂Γ

∂

( )

C

C . It is worth noting that all these quantities are

taken constant and equal to their equilibrium value for a spherical droplet at rest.

The linear oscillations of the droplet at mode n involve 13 physical parameters mentioned above and therefore depend on 10 independent dimensionless groups: the inner Reynolds numbers, Rei=ρia2ωL/ηi(or the outer one, Reo=ρoa2ωL/ηo);

the bulk Péclet number, Pe = a2_ω

L/D; the surface Péclet

number, PeS= a2ωL/DS; the density ratio,ρ̂ = ρo/ρi; the viscosity

ratio,η̂ = η_o/η_i; the three normalized elasticities, E_Gibbs* = E_Gibbs/ γe, Ed* = Ed/γe, and Es* = Es/γe; the two normalized surface

viscosities,η_d*=η_d/aη_iandη_s*=η_s/aη_i; the dimensional group that compares the diﬀusion time scale to the oscillation period, λ = D1/2_/_Λ

CωL1/2. In these expressions,ωLis the Lamb frequency,

which corresponds to the frequency of oscillation under the assumption of potentialﬂow and pure interface

ω γ ρ ρ = − + + + + n n n n a n n ( 1) ( 1)( 2) ( ( 1) ) L 2 e 3 o i (8)

In spherical coordinates, the location of the droplet interface is deﬁned by its complex local radius r̃as a function of polar angle θ and the azimuthal angle ϕ. The local radius r̃ is written, for mode (n,m), as θ ϕ = ̃ − + ϵ θ ϕ −α ω = F( , ) r a(1 n m n m, Y, ( , )e t) 0 2 L (9)

where F is the surface equation, Yn,mare the spherical harmonics,

ϵn,mis the normalized mode amplitude, andα2= iωn/ωL+βn/ωL

is the normalized complex frequency. Modes of the same n but

diﬀerent m are degenerated, which means that they have the

same α2. For that reason, we will further consider only

axisymmetric modes, which correspond to m = 0 (i.e., with a normalized amplitudeϵn=ϵn,0) and which do not depend on the

azimuthal coordinateϕ.

We now present the problem equations, which have been linearized by assuming small amplitude oscillations (ϵn≪ 1). In

the following, all variables are made dimensionless: lengths are normalized by a, times by ω_L−1, surfactant surface concen-trations byΓ0, surfactant volume concentrations byΓ0/ΛC(the

problem being linear inΓ₀, we can setΓ₀= 1 without loss of generality).

Dynamic Equations for the Fluids. The dynamics of each of the twoﬂuids is driven by the continuity and the Navier−Stokes equations. In the following, subscripts i and o denote the inner and the outer phases, respectively. The continuity equation is

∇· =v 0 (10)